Rationalising Denominators Flashcards

All 8 cards in this deck

What does 'rationalise the denominator' mean?

Rewrite the fraction as an equal fraction whose denominator is a rational number (no surd in it).

What are the steps to rationalise a denominator of the form $\frac{k}{\sqrt{a}}$?

e.g. $\frac{2}{\sqrt{11}}$

1. Multiply numerator and denominator by $\sqrt{11}$
2. Denominator: $\sqrt{11}\times\sqrt{11}=11$
3. Answer: $\frac{2\sqrt{11}}{11}$

What are the steps to rationalise a denominator of the form $\frac{k}{a+\sqrt{b}}$?

e.g. $\frac{14}{2+\sqrt{3}}$

1. Multiply numerator and denominator by the conjugate $2-\sqrt{3}$
2. Expand both: $\frac{28-14\sqrt{3}}{4-3}$
3. Simplify: $28-14\sqrt{3}$

What is $(a+\sqrt{b})(a-\sqrt{b})$ equal to?

e.g. $(2+\sqrt{3})(2-\sqrt{3})$

$a^2 - b$, which is rational (the surd terms cancel).
$(2+\sqrt{3})(2-\sqrt{3}) = 4-3 = 1$

True or false? Expanding $(2+\sqrt{3})(2-\sqrt{3})$ gives $4+3=7$

False. $\sqrt{3}\times(-\sqrt{3}) = -3$, not $+3$, so the answer is $4-3=1$.

What should you do first when writing $\frac{8+\sqrt{12}}{5+\sqrt{3}}$ in the form $\frac{a+\sqrt{3}}{b}$?

Simplify $\sqrt{12}$ to $2\sqrt{3}$ so all surds match, then multiply numerator and denominator by the conjugate $5-\sqrt{3}$.

A student rationalises $\frac{14}{2+\sqrt{3}}$ by multiplying the numerator and denominator by $2+\sqrt{3}$. What is the error?

You must multiply by the conjugate $2-\sqrt{3}$; multiplying by $2+\sqrt{3}$ leaves a surd in the denominator.

True or false? To rationalise $\frac{5}{\sqrt{12}}$ you may multiply just the denominator by $\sqrt{12}$.

False. You must multiply the numerator and the denominator by $\sqrt{12}$, or the value of the fraction changes.